An arithmetic count of the lines on a smooth cubic surface
We give an arithmetic count of the lines on a smooth cubic surface over an arbitrary field, generalizing the counts that over there are lines, and over the number of hyperbolic lines minus the number of elliptic lines is. In general, the lines are defined over a field extension and have an associated arithmetic type in. There is an equality in the Grothendieck-Witt group of, where denotes the trace. Taking the rank and signature recovers the results over and. To do this, we develop an elementary theory of the Euler number in-homotopy theory for algebraic vector bundles. We expect that further arithmetic counts generalizing enumerative results in complex and real algebraic geometry can be obtained with similar methods.
Leo Kass, J; Wickelgren, K
Start / End Page
International Standard Serial Number (ISSN)
Digital Object Identifier (DOI)